Best quadrature formula for the classW r L 2

В РАБОтЕ РЕшЕНА жАДАЧ А О НАИлУЧшЕИ кВАДРАт УРНОИ ФОРМУлЕ ВИДА \(\mathop \Sigma \limits_{\iota = 1}^n p_i f(x_i )\) НА клАс сЕ пЕРИОДИЧЕскИх ФУНкц ИИW ^r L ₂.     

Best quadrature formulas on classes of differentiable periodic functions

A solution is given to the problem of finding the best quadrature formula among formulas of the form $$\int_0^{2\pi } {f(x)dx \approx \sum\nolimits_{k = 0}^{m - 1} {\sum\nolimits_{l = 0}^\rho {pk,l} f^{(l)} (x_k ),} } $$ which are exact in the case of a constant, for p = r ? 1, r = 1, 2, 3... and p = r ? 2, r even, for the classes W(r) LqM of 2π-periodic functions.     

Best quadrature formula on Sobolev class with Chebyshev weight

Using best interpolation function based on a given function information, we present a best quadrature rule of function on Sobolev class KW^r[-1,1]${\mathit{KW}}^r[-1,1]$ with Chebyshev weight. The given function information means that the values of a function f∈KW^r[-1,1]$f\in {\mathit{KW}}^r[-1,1]$ and its derivatives up to r-1$r-1$ order at a set of nodes x$\mathbf{x}$ are given. Error bounds are obtained, and the method is illustrated by some examples.     

Best approximation of periodic functions by trigonometric polynomials in L2

Estimates are gotten for the best approximations in L₂ (0, 2) of a periodic function by trigonometric polynomials in terms of its m-th continuity modulus or in terms of the continuity modulus of its r-th derivative. The inequality E_n–1(f)L₂<(C _2m ^m _m(2/n;f)L₂(1 const) is proved, where the constant (C _2m ^m )^–1/2 is unimprovable for the whole space L₂ (0, 2). Two titles are cited in the bibliography.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 513–522, November, 1967.     

Best approximation by splines on classes of periodic functions in the metric of L

We have obtained the exact value of the upper bound on the best approximations in the metric of L on the classes W^rH of functionsf C ₂ ^r for which ¦f ^(r) (x)-f ^(r) (x)) ¦ <(¦ x-xf) [ (t) is the upwards-convex modulus of continuity] by subspaces of r-th order polynomial splines of defect 1 with respect to the partitioning k/n.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 655–664, November, 1976.     

Exact asymptotic for the remainder of Hermite's quadrature formula for function classes W p 2

For function classes W _p ² we obtain an asymptotically exact estimate of error of Hermite's quadrature formula (the quadrature formula of the highest algebraic degree of precision corresponding to a weight function .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 119–122, January, 1990.     

Best quadrature formulas for improper integrals on certain classes of differentiable functions

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 132–134, June, 1991.     

Modification of the Euler quadrature formula for functions with a boundary-layer component

The Euler quadrature formula for the numerical integration of functions with a boundary-layer component on a uniform grid is investigated. If the function under study has a rapidly growing component, the error can be significant. A uniformly accurate quadrature formula is constructed by modifying the Hermite interpolation formula so that the resulting one is exact for the boundary-layer component. An analogue of the Euler formula that is exact for the boundary-layer component is constructed. It is proved that the resulting composite quadrature formula is third-order accurate in space uniformly with respect to the boundary-layer component and its derivatives.     

On the best interval quadrature formulae for classes of differentiable periodic functions

In this paper we solve the problem about optimal interval quadrature formula for the class W^rF of differentiable periodic functions with rearrangement invariant set F of their derivatives of order r. We prove that the formula with equal coefficients and n node intervals having equidistant midpoints is optimal for considering classes. To this end a sharp inequality for antiderivatives of rearrangements of averaged monosplines is proved.     

The best quadrature formula based on Hermite information for the class KW~2[a,b]

The best quadrature formula has been found in the following sense:for afunction whose norm of the second derivative is bounded by a given constant and thebest quadrature formula for the approximate evaluation of integration of that function canminimize the worst possible error if the values of the function and its derivative at certainnodes are known.The best interpolation formula used to get the quadrature formula aboveis also found.Moreover,we compare the best quadrature formula with the open compoundcorrected trapezoidal formula by theoretical analysis and stochastic experiments.     

Best approximations of the classes B p,θ r of periodic functions of many variables in uniform metric

We obtain estimates exact in order for the best approximations of the classes B _∞,θ ^r of periodic functions of two variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes B _p,θ ^r , 1 ≤ p < ∞, of periodic functions of many variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1395–1406, October, 2006.     

Best approximation of a class of functions by another class

Let B be the space C=C(I) or L=L₁(I), where I=(–, ). By ⁽ⁿ⁾(x)_c we will mean the upper bound of the modulus of the values of the derivative of the function 相似文献   

Sobolev类带权函数的基于给定信息的最佳求积公式

给出了r阶Sobo lev类KWr[a,b]带权函数的基于给定信息的最佳求积公式和它的误差估计式.这里的给定信息是指:已知函数在给定区间若干点上的函数值和直到r-1阶导数值.对r≤2,得到了最佳求积公式和误差估计式的显式结果.另外还给出了类KW2[a,b]中在节点的导数值为零的函数所组成的子类的相应的最佳求积公式.